Chapter 7

\(2 x+y=6 \quad(-2,10),(-1,5),(3,0)\)

$$ y=-x+3 \quad \begin{array}{l|llll} x & -2 & -1 & 0 & 4 \\ \hline y & & & & \end{array} $$

$$ y=2 x-1 \quad \begin{array}{l|llll} \mathbf{x} & -3 & -1 & 0 & 2 \\ \hline \mathbf{y} & & & & \end{array} $$

\(y \leq-x+2\)

$$ \begin{array}{ll|llll} 2 x-y=6 & \mathbf{x} & -2 & 0 & 2 & 4 \\ \hline \mathbf{y} & & & & \end{array} $$

Verify that the points \((-3,1),(5,7)\), and \((8,3)\) are vertices of a right triangle. [Hint: If \(a^{2}+b^{2}=c^{2}\), then it is a right triangle with the right angle opposite side \(c\).]

Verify that the points \((0,3),(2,-3)\), and \((-4,-5)\) are vertices of an isosceles triangle.

\(-2 x+y-3 \leq 0\)

Verify that the points \((7,12)\) and \((11,18)\) divide the line segment joining \((3,6)\) and \((15,24)\) into three segments of equal length.

Verify that \((3,1)\) is the midpoint of the line segment joining \((-2,6)\) and \((8,-4)\).

\(y \geq-2\)

\(x^{2}+y^{2}-4 x-12=0\)

Why is the point \((-4,1)\) not a good test point to use when graphing \(5 x-2 y>-22\) ?

Explain how you would graph the inequality $$ -3>x-3 y . $$

Graph \(|x|<2\). [Hint: Remember that \(|x|<2\) is equivalent to \(-2

Graph \(|y|>1\).

\(2 y=x-2\)

Find \(x\) if the line through \((-2,4)\) and \((x, 6)\) has a slope of \(\frac{2}{9}\)

Graph \(|x+y|<1\).

Find \(y\) if the line through \((1, y)\) and \((4,2)\) has a slope of \(\frac{5}{3}\)

Graph \(|x-y|>2\).

Find \(x\) if the line through \((x, 4)\) and \((2,-5)\) has a slope of \(-\frac{9}{4}\).

Find \(y\) if the line through \((5,2)\) and \((-3, y)\) has a slope of \(-\frac{7}{8}\)

Use the DRAW feature of your graphing calculator to draw each of the following. (a) A line segment between \((-2,-4)\) and \((-2,5)\) (b) A line segment between \((2,2)\) and \((5,2)\) (c) A line segment between \((2,3)\) and \((5,7)\) (d) A triangle with vertices at \((1,-2),(3,4)\), and \((-3,6)\)

\(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

\(x\) intercept of \(-3\) and slope of \(-\frac{5}{8}\)

\(x\) intercept of 5 and slope of \(-\frac{3}{10}\)

Contains the point \((2,-4)\) and is parallel to the \(y\) axis

Contains the point \((-3,-7)\) and is parallel to the \(x\) axis

Contains the point \((5,6)\) and is perpendicular to the \(y\) axis

Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis

Contains the point \((1,3)\) and is parallel to the line \(x+5 y=9\)

(a) Digital Solutions charges for help-desk services according to the equation \(c=0.25 m+10\), where \(c\) represents the cost in dollars and \(m\) represents the minutes of service. Complete the following table. \(\begin{tabular}{l|llllll} \)\boldsymbol{m}\( & 5 & 10 & 15 & 20 & 30 & 60 \\ \hline \)\boldsymbol{c}\( & & & & & & \end{tabular}\)(b) Label the horizontal axis \(m\) and the vertical axis \(c\), and graph the equation \(c=0.25 m+10\) for nonnegative values of \(\mathrm{m}\). (c) Use the graph from part (b) to approximate values for \(c\) when \(m=25,40\), and 45 . (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(c=0.25 m+10\).

\(y=-x^{3}\)

(a) The equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\) can be used to convert from degrees Celsius to degrees Fahrenheit. Complete the following table. \begin{tabular}{l|llllllllll} \(\mathbf{C}\) & 0 & 5 & 10 & 15 & 20 & \(-5\) & \(-10\) & \(-15\) & \(-20\) & \(-25\) \\ \hline \(\mathbf{F}\) & & & & & & & \end{tabular} (b) Graph the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\). (c) Use your graph from part (b) to approximate values for \(\mathrm{F}\) when \(\mathrm{C}=25^{\circ}, 30^{\circ},-30^{\circ}\), and \(-40^{\circ}\). (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\).

Contains the origin and is parallel to the line \(4 x-7 y=3\)

(a) A doctor's office wants to chart and graph the linear relationship between the hemoglobin Alc reading and the average blood glucose level. The equation \(G=30 h-60\) describes the relationship, in which \(h\) is the hemoglobin Alc reading and \(G\) is the average blood glucose reading. Complete this chart of values: \begin{tabular}{l|lllllll} Hemoglobin A1c, \(\boldsymbol{h}\) & \(6.0\) & \(6.5\) & \(7.0\) & \(8.0\) & \(8.5\) & \(9.0\) & \(10.0\) \\ \hline Blood glucose, \(\boldsymbol{G}\) & & & & & & & \end{tabular} (b) Label the horizontal axis \(h\) and the vertical axis \(G\), then graph the equation \(G=30 h-60\) for \(h\) values between \(4.0\) and \(12.0\). (c) Use the graph from part (b) to approximate values for \(G\) when \(h=5.5\) and 7.5. (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(G=30 h-60\).

Contains the origin and is parallel to the line \(-2 x-9 y=4\)

Suppose that the daily profit from an ice cream stand is given by the equation \(p=2 n-4\), where \(n\) represents the gallons of ice cream mix used in a day and \(p\) represents the dollars of profit. Label the horizontal axis \(n\) and the vertical axis \(p\), and graph the equation \(p=2 n-4\) for nonnegative values of \(n\).

Contains the point \((-1,3)\) and is perpendicular to the line \(2 x-y=4\)

The cost (c) of playing an online computer game for a time \((t)\) in hours is given by the equation \(c=3 t+5\). Label the horizontal axis \(t\) and the vertical axis \(c\), and graph the equation for nonnegative values of \(t\).

Contains the point \((-2,-3)\) and is perpendicular to the line \(x+4 y=6\)

The area of a sidewalk whose width is fixed at 3 feet can be given by the equation \(A=3 l\), where \(A\) represents the area in square feet and \(l\) represents the length in feet. Label the horizontal axis \(l\) and the vertical axis \(A\), and graph the equation \(A=3 l\) for nonnegative values of \(l\).

Is perpendicular to the line \(-2 x+3 y=8\) and contains the origin.

An online grocery store charges for delivery based on the equation \(C=0.30 p\), where \(C\) represents the cost of delivery in dollars and \(p\) represents the weight of the groceries in pounds. Label the horizontal axis \(p\) and the vertical axis \(C\), and graph the equation \(C=0.30 p\) for nonnegative values of \(p\).

How do we know that the graph of \(y=-3 x\) is a straight line that contains the origin?

How do we know that the graphs of \(2 x-3 y=6\) and \(-2 x+3 y=-6\) are the same line?

What is the graph of the conjunction \(x=2\) and \(y=4\) ? What is the graph of the disjunction \(x=2\) or \(y=4\) ? Explain your answers.

Your friend claims that the graph of the equation \(x=2\) is the point \((2,0)\). How do you react to this claim?

\(|x-y|=4\)